Science:Math Exam Resources/Courses/MATH104/December 2014/Question 01 (l)

MATH104 December 2014

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Question 01 (l)
Suppose $f$ is differentiable on $(-\infty ,\infty )$ and assume it has a local extreme value at the point $x=2$ where $f(2)=0$. Let $h(x)=xf(x)+2x+3$ for all values of $x$. Does $h(x)$ have a local extreme value at $x=2$?

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Hint
If $h$ has an extreme value at some $x_{0}$, what does that tell you about $h(x_{0})$?

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. If $h(x)$ has a local extreme value at $x=2$ then $h'(2)=0$. Since $f$ has a local extrema at $x=2$, we have that $f'(2)=0$. First let us find $h'(x)$ by using the product rule: $h'(x)=f(x)+xf'(x)+2$ Substituting in $x=2$ we obtain that $h'(2)=f(2)+2f'(2)+2=0+2(0)+2=2\neq 0$. Since $h'(2)\neq 0$, $h$ does not have a local extrema at $x=2$.

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